Answer:
Step-by-step explanation:
Given that
[tex]r(t) = (t,t,e^t)[/tex]
To find tangent, normal and binormal vectors at (0,0,1)
i) Tangent vector
[tex]r'(t) = (1,1,e^t)\\[/tex]
At the particular point, r'(t) = (1,1,e)
Tangent vector = [tex]\frac{r'(t)}{||r'(t)||} \\=\frac{(1,1,e)}{\sqrt{1+1+e^2} } \\=\frac{(1,1,e)}{\sqrt{2+e^2} }[/tex]
ii) Normal vector
T'(t) = [tex](0,0,e^t)\\[/tex]
At that point T'(t) = (0,0,e)/e = (0,0,1)
iii) Binormal
B(t) = TX N
= [tex]\left[\begin{array}{ccc}i&j&k\\1&1&e^t\\0&0&e^t\end{array}\right] \\= e^t(i-j)[/tex]
